The gravitational Euler-Poisson system provides a basic model of a self-gravitating isolated star. The simplest nontrivial solutions are the radially symmetric steady states known as Lane-Emden stars - some of their key features are the spatial inhomogeneity of the associated fluid density and the presence of vacuum boundaries, either at finite radii or at infinity. The rigorous description of the phase space around them is a challenging open question. I will provide an overview of known results and then explain a new instability mechanism of accelerated shock formation, which applies to perturbations of Lane-Emden stars for the energy-critical value of the polytropic pressure law.  This is a joint work with Juhi Jang, Sung-Jin Oh, and Ely Sandine.